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Second Derivative in Matlab

  1. Jul 17, 2016 #1
    I have a set of data as follows, How can I calculate the second derivative of the curve obtained from these data.

    x=[0.1;0.07;0.05;0.03;0]; r=[-98.9407;-105.7183;-111.2423;-116.0320;-120.0462];
     
  2. jcsd
  3. Jul 17, 2016 #2
    How about fitting it to a curve, then calculating the second derivative analytically from the fitted curve formula? Like this:

    Code (Text):
    x=[0.1;0.07;0.05;0.03;0];
    r=[-98.9407;-105.7183;-111.2423;-116.0320;-120.0462];
    plot(x, r, 'r*-', 'LineWidth', 2);
    % Fit to a cubic
    coefficients = polyfit(x, r, 4)
    xFit = linspace(min(x), max(x), 100);
    rFit = polyval(coefficients, xFit);
    hold on;
    plot(xFit, rFit, 'b-', 'LineWidth', 2);
    grid on;
    legend('Actual', 'fit');

    % Set up figure properties:
    % Enlarge figure to full screen.
    set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
    % Get rid of tool bar and pulldown menus that are along top of figure.
    set(gcf, 'Toolbar', 'none', 'Menu', 'none');
    % Give a name to the title bar.
    set(gcf, 'Name', 'Demo by ImageAnalyst', 'NumberTitle', 'Off')

    % Equation is a1*x^4 + a2*x^3 + a3*x^2 + a4*x + a5
    % First derivative is 4*a1*x^3 + 3*a2*x^2 + 2*a3*x + a4
    % Second derivative is 12*a1*x^2 + 6*a2*x + 2*a3
    a1 = coefficients(1)
    a2 = coefficients(2)
    a3 = coefficients(3)
    deriv2 = 12*a1*xFit.^2 + 6*a2*xFit + 2*a3
     
     
  4. Jul 18, 2016 #3
    Thanks.
    But how to calculate second derivative graph x on r?
     
  5. Jul 18, 2016 #4
    What does that mean? I showed you how to calculate the second derivative of r with respect to x. Is that not what you want? Is r the independent variable instead of x? If so, just swap x and r in my code. Or do you just need to know how to call plot()???

    Code (Text):

    plot(xFit, deriv2, 'b-');
     
     
  6. Jul 18, 2016 #5
  7. Jul 18, 2016 #6
    You can implement this method with conv(), though with such a small array as the poster posted, the edge effects will reduce the valid region to about one element. However, for a longer vector (more elements), conv() is a good way to take the numerical/empirical derivative (vs. the analytical derivative as in my answer).
     
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